# Full text of "Treatise On Analysis Vol-Ii"

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```21    PRODUCT OF MEASURES       229

is integrable, by virtue of (13.21.14), hence
f®9 is locally v-integrable (13.13.1). Also, for each we Jfc(X) and each
v e jJfc(Y), we have

[ \f(x)g(y)u(x)v(y) dl(x) du(y) = M. f(x)u(x) d*(x)j ( (g(y)v(y) d^(y}\
by virtue of (1 3.21 .1 4) ; hence the formula (1 3.21 .1 6.1 ) follows from (1 3.21 .1 ).
(13.21.17)   Let  A, IJL be complex measures   on X,  Y,  respectively.  Then

We may write A =/• |A| and // = g • \u\, where |/| = \g\ = \ (13.16.3).
Hence \f®g\ - 1, and the result therefore follows from (13.21.16) and
(13.13.4).

(13.21.18) Let X be a complex measure on X and in a complex measure on Y.
(i) Ifl is concentrated on A cz X and p, is concentrated on B c Y (13.18),
then A ® \JL is concentrated on A x B.

(ii)   Supp(l <g) u) = Supp(/l) x Supp(/x).

(iii)    With the product convention of (13.11) we have

(13.21.18.1)

In particular, if X and ILL are bounded, then so is X ® u.

By virtue of (1 3.21 .1 7) we may restrict ourselves to the case in which /I and
// are positive. Then (i) follows from the fact that X x Y — A x B is the
union of the (A ® ^-negligible sets (X - A) x Yand X x (Y - B) (13.21.12).
It follows from (i) that Supp(A ® u) c Supp(Jl) x Supp(/i).

On the other hand, if x e Supp(A) and y E Supp(/j), then for each compact
neighborhood V of x in X and each compact neighborhood W of y in Y, we
have A(V)>0 and ^(W) > 0, whence by (13.21.15) (A®M)(VxW) =
l(V)/i(W) > 0. This establishes (ii), by virtue of the definition of neighborhoods
in X x Y. Finally, (iii) follows from (13.21.11) applied to/= (px and# = <pY,
except when one of the factors on the right-hand side is 0 and the other
is +00; but in this case we have A® ^ = 0 by (13.21.12), and therefore the
formula (13.21.18.1) remains valid in this case.

(13.21.19)   There are analogous definitions and results for the product of any
finite number of measures : if (X^ ^ ^ „ is any finite sequence of locally compactgligible
```